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Properties

Label 2-20e2-25.6-c1-0-3
Degree 22
Conductor 400400
Sign 0.535−0.844i0.535 - 0.844i
Analytic cond. 3.194013.19401
Root an. cond. 1.787181.78718
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.690 + 2.12i)5-s − 0.618·7-s + (−0.618 + 1.90i)9-s + (1.61 + 4.97i)11-s + (0.572 − 1.76i)13-s + (0.690 + 2.12i)15-s + (4.23 + 3.07i)17-s + (0.690 + 0.502i)19-s + (−0.500 + 0.363i)21-s + (−1.16 − 3.57i)23-s + (−4.04 − 2.93i)25-s + (1.54 + 4.75i)27-s + (2.92 − 2.12i)29-s + (−2.42 − 1.76i)31-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + (−0.309 + 0.951i)5-s − 0.233·7-s + (−0.206 + 0.634i)9-s + (0.487 + 1.50i)11-s + (0.158 − 0.489i)13-s + (0.178 + 0.549i)15-s + (1.02 + 0.746i)17-s + (0.158 + 0.115i)19-s + (−0.109 + 0.0792i)21-s + (−0.242 − 0.746i)23-s + (−0.809 − 0.587i)25-s + (0.297 + 0.915i)27-s + (0.543 − 0.394i)29-s + (−0.435 − 0.316i)31-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.535−0.844i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((0.535−0.844i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.535−0.844i0.535 - 0.844i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(81,⋅)\chi_{400} (81, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), 0.535−0.844i)(2,\ 400,\ (\ :1/2),\ 0.535 - 0.844i)

Particular Values

L(1)L(1) ≈\approx 1.26113+0.693313i1.26113 + 0.693313i
L(12)L(\frac12) ≈\approx 1.26113+0.693313i1.26113 + 0.693313i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(0.690−2.12i)T 1 + (0.690 - 2.12i)T
good3 1+(−0.809+0.587i)T+(0.927−2.85i)T2 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2}
7 1+0.618T+7T2 1 + 0.618T + 7T^{2}
11 1+(−1.61−4.97i)T+(−8.89+6.46i)T2 1 + (-1.61 - 4.97i)T + (-8.89 + 6.46i)T^{2}
13 1+(−0.572+1.76i)T+(−10.5−7.64i)T2 1 + (-0.572 + 1.76i)T + (-10.5 - 7.64i)T^{2}
17 1+(−4.23−3.07i)T+(5.25+16.1i)T2 1 + (-4.23 - 3.07i)T + (5.25 + 16.1i)T^{2}
19 1+(−0.690−0.502i)T+(5.87+18.0i)T2 1 + (-0.690 - 0.502i)T + (5.87 + 18.0i)T^{2}
23 1+(1.16+3.57i)T+(−18.6+13.5i)T2 1 + (1.16 + 3.57i)T + (-18.6 + 13.5i)T^{2}
29 1+(−2.92+2.12i)T+(8.96−27.5i)T2 1 + (-2.92 + 2.12i)T + (8.96 - 27.5i)T^{2}
31 1+(2.42+1.76i)T+(9.57+29.4i)T2 1 + (2.42 + 1.76i)T + (9.57 + 29.4i)T^{2}
37 1+(0.0729−0.224i)T+(−29.9−21.7i)T2 1 + (0.0729 - 0.224i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.236−0.726i)T+(−33.1−24.0i)T2 1 + (0.236 - 0.726i)T + (-33.1 - 24.0i)T^{2}
43 1−4.85T+43T2 1 - 4.85T + 43T^{2}
47 1+(−0.5+0.363i)T+(14.5−44.6i)T2 1 + (-0.5 + 0.363i)T + (14.5 - 44.6i)T^{2}
53 1+(−2.80+2.04i)T+(16.3−50.4i)T2 1 + (-2.80 + 2.04i)T + (16.3 - 50.4i)T^{2}
59 1+(−3.35+10.3i)T+(−47.7−34.6i)T2 1 + (-3.35 + 10.3i)T + (-47.7 - 34.6i)T^{2}
61 1+(−2.69−8.28i)T+(−49.3+35.8i)T2 1 + (-2.69 - 8.28i)T + (-49.3 + 35.8i)T^{2}
67 1+(−3.85−2.80i)T+(20.7+63.7i)T2 1 + (-3.85 - 2.80i)T + (20.7 + 63.7i)T^{2}
71 1+(5.35−3.88i)T+(21.9−67.5i)T2 1 + (5.35 - 3.88i)T + (21.9 - 67.5i)T^{2}
73 1+(2.78+8.55i)T+(−59.0+42.9i)T2 1 + (2.78 + 8.55i)T + (-59.0 + 42.9i)T^{2}
79 1+(6.54−4.75i)T+(24.4−75.1i)T2 1 + (6.54 - 4.75i)T + (24.4 - 75.1i)T^{2}
83 1+(5.04+3.66i)T+(25.6+78.9i)T2 1 + (5.04 + 3.66i)T + (25.6 + 78.9i)T^{2}
89 1+(2.76+8.50i)T+(−72.0+52.3i)T2 1 + (2.76 + 8.50i)T + (-72.0 + 52.3i)T^{2}
97 1+(−3.11+2.26i)T+(29.9−92.2i)T2 1 + (-3.11 + 2.26i)T + (29.9 - 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.41535371494637226609520112039, −10.35490414906394033419346201061, −9.843748913864362287029030510654, −8.444063948624921235690630365980, −7.64964169634672013110252444161, −6.93661553329277548249277608163, −5.81593469294710744862185196303, −4.35483719039384631653525408076, −3.14731942388278701332579685067, −1.98819394678751910442890975151, 0.971155834084136433971578650085, 3.17170174766898298690705606322, 3.93613951698086123769483032382, 5.29736920317658882506503356110, 6.24782377779695983774672907165, 7.59449742170499026996537561661, 8.675984357603885905740269507592, 9.090227335867207349068232064586, 9.975418250189045427373611926248, 11.37928188866988108406516717824

Graph of the ZZ-function along the critical line